Accelerated convergence in TLM algorithms for the Laplace equation

Abstract

Transmission line matrix (TLM) schemes for the Laplace equation exhibit some curious features. There exist values of TLM parameters where convergence towards the analytical values is very rapid. This phenomenon is first examined using a binary scattering approach. A Fourier analysis of the equivalently bounded diffusion equation does not reveal any features that would account for these observations. However, a similar analysis using the Telegraphers' equation suggests that a TLM model under optimum conditions is operating at the transition between real and imaginary solutions. Small differences between the optimized parameters predicted by the two approaches are probably due to inaccuracies in the Fourier description of the heat-injecting boundary condition that is used in TLM models

A transmission line modelling (TLM) method for steady‐state convection–diffusion

“…For any model, the number of time steps required to satisfy the convergence condition drops as expected as t increases but then rises again above an optimum time step length. This behaviour has been noted previously in TLM diffusion models [11]. The increase is associated with the wave-propagation term in the differential equation for the lossy TL which becomes significant at higher t values and leads to wave-like behaviour which takes time to settle down.…”

“…, and R d (x) satisfy Equations (10), (7), and (8), then the fourth term in Equation (2) becomes (10) and (11) gives the distributed inductance in terms of u…”

“…It has been shown that, because it is stable with long time steps, lossy TLM can rapidly converge to a steady-state solution [11]. For any particular problem, there exist optimum values for the TLM parameters that result in ultra-fast convergence.…”

An alternative TLM method for steady‐state convection–diffusion

“…the length of the TL) and time are divided into finite increments. Traditionally steady-state solutions have been found by running the scheme until transients reduce to an acceptable level [7,10,11], but a recent paper has shown that they can also be found directly [5]. The first step in modelling a transmission line using TLM is to choose the locations of the nodes at which the solution will be calculated and a time step length, Δt.…”

“…The difference between the instantaneous node voltage and the incident voltages leads to pulses being scattered from the node, one to the left (10) and one to the right…”